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Theorem sspwimpALT2 43992
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpALT2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3477 . . . 4 𝑥 ∈ V
2 elpwi 4609 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 id 22 . . . . 5 (𝐴𝐵𝐴𝐵)
42, 3sylan9ssr 3996 . . . 4 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
5 elpwg 4605 . . . . 5 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
65biimpar 477 . . . 4 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
71, 4, 6sylancr 586 . . 3 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
87ex 412 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
98ssrdv 3988 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  Vcvv 3473  wss 3948  𝒫 cpw 4602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-pw 4604
This theorem is referenced by: (None)
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